1. Field of the Invention
The present invention relates generally to satellites and more particularly to satellite orbits.
2. Description of the Related Art
The orbit diagram 20 of FIG. 1 illustrates relationships between movements of a satellite, the earth, the sun and the moon. In the diagram, a satellite 22 orbits the earth 24 in an orbit plane 26. An orbit axis that is orthogonal to and centered in the orbit plane is typically referred to as an orbit normal 28. Similarly, the earth rotates about an equatorial pole 32 which is normal to an equatorial plane 30.
The sun appears to move about the earth 24 in an ecliptic plane 36 which has an ecliptic pole 38 that is tilted .about.23.44 degrees from the equatorial pole 32. The sun 34 is shown at two exemplary positions, a winter solstice position 34W and a summer solstice position 34S. Similarly, the moon 40 is shown at two positions 40A and 40B which respectively represent its farthest and closest approach to the equatorial plane 30.
The moon orbits the earth in a lunar orbit plane 42 which has a lunar orbit normal 44. The lunar orbit normal 44 is canted .about.5.14 degrees from the ecliptic pole 38 and regresses (i.e., rotates clockwise as viewed from north of the ecliptic plane 36) about that pole with a period of .about.18.6 years. This regression is indicated by movement arrow 45 and the lunar orbit normal 44 is shown in two extreme positions 44A and 44B that correspond to extreme orientations 42A and 42B of the lunar orbit plane 42.
The celestial diagram 48 of FIG. 2 relates the orbit plane 26 and the equatorial plane 30 to an equatorial coordinate system 50 (the plane 26 is shown at a greater angle in FIG. 2 for clarity of illustration). The system 50 has 3 orthogonal axes; an e.sub.3 axis coaxial with the equatorial pole (32 in FIG. 1), an axis e.sub.1 that is oriented through the vernal equinox (and points generally to Aries) and a third orthogonal axis e.sub.2. The right ascension of a celestial body is its angle (taken counterclockwise from above the e.sub.1 -e.sub.2 plane) from the axis e.sub.1 and the declination of a celestial body is its angle from the e.sub.1 -e.sub.2 plane (alternatively, the body's codeclination is its angle from the e.sub.3 axis).
In reference to the equatorial plane 30, the orbit plane 26 has an ascending node 52 where the satellite 22 crosses to the upper side of the equatorial plane and a descending node 54 where it crosses to the lower side (satellite motion is indicated by movement arrow 55). With respect to the coordinate system 50, an orbit plane's inertial position is typically specified by its inclination I and its right ascension of the ascending node (RAAN). In FIG. 2, the inclination I is the angle 56 between the equatorial plane 30 and the orbit plane 26 and the RAAN is the angle 58.
The earth, the sun and the moon all perturb a satellite's orbit plane. The dominant sources of these orbit-plane perturbations are the oblateness (i.e., polar flattening) of the earth and the gravitational-gradients generated by the sun and the moon. Various references (e.g., R. R. Allan and G. E. Cook, "The long-period motion of the plane of a distant circular orbit", Proceedings Royal Society, 1964, vol 280, pp. 97-109) have shown that these perturbations cause an orbit normal to regress about a theoretical vector that is hereinafter referred to as a Q vector.
The orbit normal 28 and a Q vector 60 are shown in the diagram 61 of FIG. 3 with reference to the equatorial coordinate system 50 of FIG. 2. The Q vector 60 has three vector components; a first fixed vector 62 associated with earth-induced perturbations and oriented coaxially with the e.sub.3 axis, a second fixed vector 64 associated with sun-induced perturbations and oriented parallel to the ecliptic pole (38 in FIG. 1) and a rotating vector 66 associated with moon-induced perturbations. The latter vector is parallel to the lunar orbit normal (44 in FIG. 1) and, accordingly, it cones about the vector component 64 at an angle of .about.5.14 degrees and with a period of .about.18.6 years (for clarity of illustration, the angle is considerably increased in FIG. 3). As shown in FIG. 3, the resultant Q vector 60 tilts by an offset angle .phi. from the e.sub.3 axis and the equatorial pole (32 in FIG. 1).
If the orbit normal 28 is initially tilted from the Q vector 60 by an angle .theta., the combined perturbations cause the orbit normal to regress, as illustrated by the motion arrow 67, about the Q vector 60 with a constant angle .theta. and at an angular rate equal to cosine .theta. times the magnitude of the Q vector (this rate is also shown in FIG. 3). As indicated by the motion arrow 68, the Q vector 60 repetitively traces a path with an .about.18.6 period and, accordingly, its magnitude and direction vary over that period. The magnitudes of the vector components 62, 64 and 66 are functions of the semimajor axis of the satellite's orbit and the satellite's mean motion (i.e., a function of the satellite's altitude).
Because of the offset angle .phi., regression of the orbit normal 28 about the Q vector 60 effects changes in the orbit plane's inclination (56 in FIG. 2). Satellite missions generally require that the inclination be controlled within a specified band. If correction is required, inclination control is typically effected through the application of corrective thruster forces to the satellite. In particular, inclination corrections are accomplished with thruster forces that are directed normal to the orbit plane and inclination corrections are the dominant users of thruster fuel. Because reduction of fuel usage permits an increase of satellite payload and revenue, extensive efforts have been directed to the reduction or elimination of thruster fuel use in control of orbit plane inclination.
Inclination control is particularly important in systems (e.g., communication systems) that require a constellation of satellites to occupy a single orbit plane. Significant savings in thruster fuel and operational complexity can be realized in these systems if the satellites can be inserted into orbit so that the need for inclination control is reduced over the system's lifetime.